05708803

j

05708803 Chatterjee, Shirshendu Durrett, Rick Contact processes on random graphs with power law degree distributions have critical value 0. Ann. Probab. 37, No. 6, 2332-2356 (2009). 2009 Institute of Mathematical Statistics, Hayward, CA EN contact process power-law random graph epidemic threshold

doi:10.1214/09-AOP471

The authors consider the contact process with infection rate $\lambda$ on a random graph with $n$ vertices and power law degree distributions. Based on mean field calculations, physicists seem to regard as an established fact that the critical value $\lambda _c$ of the infection rate is positive if the power $\alpha$ is larger than three; indeed, this result has recently been generalized to bipartite graphs by {\it J. G\'omez-Garde\~nes} et al. [Proc. Natl. Acad. Sci. USA 105, 1399--1404 (2008)]. However, it is shown in the present paper that the critical value $\lambda _c$ is zero for any value of $\alpha>3$. In addition, the contact process starting from all vertices infected, with a probability tending to one as $n$ goes to infinity, maintains a positive density of infected sites for time at least exp$(n^{1-\delta})$ for any $\delta>0$. Thanks to the last result and the contact process duality, the authors also establish the existence of a quasi-stationary distribution in which a randomly chosen vertex is occupied with probability $\rho (\lambda)$. One expects that $\rho(\lambda)\sim C\lambda^\beta $ as $\lambda$ tends to zero. The authors show that $\alpha-1\leq\beta\leq 2\alpha-3$, so that in particular $\beta$ is larger than two for $\alpha$ lager than three. As a result, even if the graph is locally tree-like, $\beta$ does not take the mean field critical value $\beta=1$. Arnaud Durand (Paris)